Persistence of multi-dimensional degenerate hyperbolic lower dimensional invariant tori in reversible systems
type: Item
instance of: scholarly article
title: Persistence of multi-dimensional degenerate hyperbolic lower dimensional invariant tori in reversible systems
author: Junxiang Xu
author: Xiao-Mei Yang
published in: Journal of Differential Equations
publication date: 2022-12-28T00:00:00Z
review text: The authors consider reversible systems with multi-dimensional degenerate hyperbolic lower dimensional tori of prescribed frequencies that persist. They focus on the reversible system given by \(\dot{w} = N(w) + (0, 0, g(v), 0) + f(w)) \) where \(w = (x, y, u, v) \in \mathbb{T}^n \times \mathbb{R}^n \times \mathbb{R}^p \times \mathbb{R}^p\), and \(N = (\omega_0 + My + Dv, 0, Cy + Av, Bu)\) where \(M, D, C, A\), and \(B\) are constant matrices. Further, \(g(v) = (v_1^{2d_1+1}, v_2^{2d_2 + 1}, \dots, v_p^{2d_p+1})\) is a higher order term, \(v = (v_1, v_2, \dots, v_p)\), and \(d_1, d_2, \dots, d_p\) are positive integers. The function \(f = (f_1, f_2, f_3, f_4)\) is a small perturbation. Two additional assumptions are made: the matrix \(M\) is nonsingular and a hyperbolicity assumption holds true, so that all the eigenvalues of the matrix \[\begin{pmatrix} 0 & A \\ B & 0 \end{pmatrix}\] have non-zero real parts. The main theorem states that if \(\omega_0\) satisfies a certain Diophantine condition and \(f\) is real analytic on a small disk about the origin with Fourier series norm that be made arbitrarily small on that disk, then the reversible system admits a lower dimensional invariant torus with prescribed frequency \(\omega_0\). The proof uses KAM theory with parameters as well as some topological degree theory.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J25
Mathematics Subject Classification ID: 37J06
Mathematics Subject Classification ID: 70H08
Mathematics Subject Classification ID: 70K43
zbMATH DE Number: 7637390
zbMATH Keywords: reversible system
zbMATH Keywords: KAM iteration
zbMATH Keywords: degenerate equilibrium point
zbMATH Keywords: lower dimensional invariant tori
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: j.jde.2022.11.037
OpenAlex ID: W4310117577
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DOI: 10.1016/J.JDE.2022.11.037
description: scientific article; zbMATH DE number 7637390